1/(1*2)+1/(2*3)+1/(3*4)+......+1/(99*100)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+……+(1/99+1/100)
=1-1/2+1/2-1/3+1/3-1/4+……+1/99-1/100
=1-1/100
=99/100